Description: The South Pacific Convergence Zone (SPCZ) and South Atlantic Convergence Zone (SACZ) are diagonal bands of precipitation that extend from the equator southeastward into the Southern Hemisphere over the western Pacific and Atlantic Oceans, respectively. With mean precipitation rates over 5 mm day −1 , they are a major component of the tropical and global climate in austral summer. However, their basic formation mechanism is not fully understood. Here, a conceptual framework for the diagonal convergence zones is developed, based on calculations of the vorticity budget from reanalysis and Rossby wave theory. Wave trains propagate eastward along the Southern Hemisphere subtropical jet, with initially quasi-circular vorticity centres. In the zonally sheared environment on the equatorward flank of the jet, these vorticity centres become elongated and develop a northwest-southeast tilt. Ray tracing diagnostics in a non-divergent, barotropic Rossby wave framework then explain the observed equatorward propagation of these diagonal vorticity structures toward the westerly ducts over the equatorial Pacific and Atlantic. The baroclinic component of these circulations leads to destabilisation and ascent ahead of the cyclonic vorticity anomaly in the wave, triggering deep convection because of the high sea surface temperatures in this region. Latent heat release then forces additional ascent and strong upper-tropospheric divergence, with an associated anticyclonic vorticity tendency. A vorticity budget shows that this cancels out the advective cyclonic vorticity tendency in the wave train over the SPCZ, and dissipates the wave within a day. The mean SPCZ is consequently comprised of the sum of these pulses of diagonal bands of precipitation. Similar mechanisms also operate in the SACZ. However, the vorticity anomalies in the wave trains are stronger, and the precipitation and negative feedback from the divergence and anticyclonic vorticity tendency are weaker, resulting in continued propagation of the wave and a more diffuse diagonal convergence zone.